Feeling lazy, so I just copied what I wrote 4 years ago in reply to a similar question:
We loved our 5 weeks in an apartment on Borgo la Croce near Piazza Beccaria. Pedestrian-only street lined with coffee bars, restaurants, and regular stores like hardware stores. Just inside the old city walls. Quiet.
A couple minutes walk west to the Sant' Ambrogio market and a couple minutes walk east to a pretty large and good supermarket. One minute walk to the bus stop for the little electric buses that can take you almost everywhere in the historic district. One and a half minute walk to what, after much testing, we thought was the best gelato in Florence, Gelateria De' Medici. About 15 minutes walk to the Duomo.
The helpful and English-speaking airbnb superhost has two apartments in the same building, one on the top floor, Le Rose Siberine (ours) and one on the floor above street-level, Le Rose di Leonardo. She lives in the building, too. We would stay in either very happily.
We have also stayed in the Oltrarno for a week and enjoyed that, too, but it was considerably busier and noisier. Of course, it depends on the exact apartment you choose. If we stayed across the river again, we'd probably stay in the San Frediano neighborhood because several of our favorite places to eat were there and it would be near the British Institute where I like to take history of art classes.
